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complex numbers rather than real numbers, as we have restricted ourselves
thus far. Using these ideas, we discuss, in a very preliminary fashion, Fourier
series and Legendre polynomials.
We use the Dirac notation to stress the commonalties that unite the finite-
and infinite-dimensional vector spaces. We, at this level, sacrifice the mathe-
matical rigor for the sake of simplicity, and even commit a few sins in our
treatment of limits. A more formal and rigorous treatment of this subject can
be found in many books on functional analysis, to which we refer the inter-
ested reader for further details.
A Hilbert space is much the same type of mathematical object as the vector
spaces that you have been introduced to in the preceding sections of this
chapter. Its elements are functions, instead of n-dimensional vectors. It is infi-
nite-dimensional because the function has a value, say a component, at each
point in space, and space is continuous with an infinite number of points.
The Hilbert space has the following properties:

1. The space is linear under the two conditions that:
a. If a is a constant and ϕ is any element in the space, then a ψ
is also an element of the space; and
b. If a and b are constants, and ϕ and ψ are elements belonging
to the space, then a ϕ + b ψ is also an element of the space.
2. There is an inner (dot) product for any two elements in the space.
The definition adopted here for this inner product for functions
defined in the interval t min ≤ t ≤ t max is:

ψϕ = ∫ t max ψ()t ϕ()t dt (7.64)
t min

3. Any element of the space has a norm (“length”) that is positive
and related to the inner product as follows:

ϕ 2 = ϕ ϕ = ∫ t max ϕ()t ϕ()t dt (7.65)
t min

Note that the requirement for the positivity of a norm is that
which necessitated the complex conjugation in the definition of
the bra-vector.
4. The Hilbert space is complete; or loosely speaking, the Hilbert
space contains all its limit points. This condition is too technical
and will not be further discussed here.

In this Hilbert space, we define similar concepts to those in finite-dimen-
sional vector spaces:

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